Okay.. solar to LiPo is nice and specific, and I gather you're running the ATTiny from the battery, not using it to control how the battery charges.
13.8 amp-hours though? Dang..
Someone will need to check me on this, but I'm pretty sure people have adapted the Adafruit LiPo charger:http://www.adafruit.com/products/259
to work with a solar panel supply. You can probably find links to reference projects in the Adafruit blog.
If I understand you correctly though -- you're using the LiPo to power the ATTiny, not using the ATTiny to regulate charging of the LiPo -- you pretty much have two separate circuits: a battery charger, and a battery-powered microcontroller.
LiPo charging is fussy enough that it's probably best to use a dedicated chip. I happen to use Microchip MCP73811Ts, which cost about 50c each, but other people have different preferences. They're all cheap, small, and are optimized to do all the measuring-deciding-and-controlling as efficiently as possible though.
Running a microcontroller from a battery is really a separate subject from battery charging. You'll want to look at the brownout and low-power shutdown modes the chip offers, and design accordingly. Given the Brick-o-Power you're charging though, I doubt you'll need to sweat over the micropower design principles.
If you're specifically trying to build a charger controlled by the microcontroller, look at the datasheets for dedicated charging chips to get an idea of how the LiPo charging process works. Your battery pack may also have a datasheet that tells you how that specific unit wants to be charged. Once you understand the trickle current/constant current/constant voltage dance, you'll be able to program the ATTiny to do the same thing.
When you void a product warrany, you give up your right to sue the manufacturer if something goes wrong and accept full responsibility for whatever happens next. And then you truly own the product.